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Retrospective Rating

Recall: Prospective WC Rating

2000 Q6

\(\begin{align} \text{Premium} = &\sum\limits_{i \in class} \dfrac{Payroll_i}{100} \text{Manual Rate}_i &\cdots (1)\\ &\times \: \text{Exp Mod} &\cdots (2)\\ &\times \: \text{Sch Mod} &\cdots (3)\\ &\times \: (1 - \text{Premium Discount %}) &\cdots (4) \end{align}\)

\((1)\) Manual Premium

\((1) \times (2)\) Modified Premium

\((1) \times (2) \times (3)\) Standard Premium

  • Authorized rates + E-mod + Loss constant + Min Premium

\((1) \times (2) \times (3) \times (4)\) Guaranteed Cost Premium

  • Assume no expense fee
  • Reflects lower proportion of fixed expense for larger risk
    • Usually calculated by Standard Premium tiers

Retrospective Rating: Intro

Uses insured’s loss experience during the current policy term to determine their current policy premium

Initial premium collected at the start (deposit premium) \(\Rightarrow\) Adjust based on reported loss starting 6 months after policy expires, then every 12 months after

\(\text{Premium} = \left[\text{U/W Expense & Profit xTax} + \left(1 + \dfrac{\mathrm{E}[LAE]}{\mathrm{E}[Loss]}\right) \times Loss_{actual}\right] \times \text{Tax Multiplier}\)

There’s always going to be a max and min premium or else this would defeat the purpose of risk transfer

Caps are applied to the premium amount \(\equiv\) Capping actual losses at max/min aggregate loss limits \(\Rightarrow\) Add a charge to the premium calculation to account for this

In addition to aggregate caps, can also limit losses per occurrence (Use limited losses in the retro premium calculation) \(\Rightarrow\) A separate charge to the premium calculation to account for the per occurrence limit

With both occurrence and agg limits, need to consider the overlap between the 2 charges

The starting point for retro premium is the standard premium

  • GCP only uses expected losses to calculate the premium
  • Retro premiums use a combination of expected loss (i.e., the insurance charge and savings) and actual loss to calculate the premiums

Retrospective Rating: w/o Occurence Limit

Important Formulas

NCCI retro premium formula for each individual policy

\(R = (b + C A + PCV) T\) Memorize

  • \(R\): Retro premium @ t \(\in [G, H]\)

  • \(b\): Basic premium = Profit + Charge for max/min premium + non-LAE expense xTax

    • Charge for max/min premium \(\equiv\) Net insurance charge
  • \(A\): Reported loss @ t (may include ALAE)

  • \(C\): Loss conversion factor \(= 1 + \dfrac{\mathrm{E}[LAE]}{\mathrm{E}[Loss]}\)

    • If \(A\) includes ALAE \(\Rightarrow\) \(= 1 + \dfrac{\mathrm{E}[ULAE]}{\mathrm{E}[Loss \& LAE]}\)
  • \(T\): Tax multiplier \(=\dfrac{0.2 + PLR(1+\mu)}{0.2 + PLR} \left({\dfrac{1}{1-\tau}}\right)\) Memorize

    • \(PLR = \text{Permissible Loss Ratio} = \dfrac{\mathrm{E}[A]}{P}\)

    • \(\mu = \text{Assessments}\)

    • \(\tau = \text{Premium Based Tax}\)

    • For an interstate risk, use a weighted average of the individual state tax multipliers with the standard premium for each state as the weights

  • \(V\): Retro development factor (Optional)

    • \(= (1-\frac{1}{CDF}) \Rightarrow \text{Unreported Losses}\)

    • \(C \times A + P \times C \times V \equiv \text{B-F Ultimate}\)

    • To stabilize premium adjustments

    • Only used for the first 3 adjustments

    • \(V\) are fixed at inception but they change over time according to the plan from the outset

    • Since losses develop over time, without a retrospective development factor, usually losses will be lower at early adjustments, which will cause refunds to the insured. Once losses develop higher, the insured will have to pay the money back. Using a retrospective development factor will reduce the back-and-forth payments, reduce the credit risk of collecting future premiums, and allow the insurer to have more premium at early adjustments to use in earning additional investment income.

  • \(P\): Standard Premium


For a balanced plan:

Expected retrospective premium = premium for a prospectively rated policy

\(b = e - (C - 1)\mathrm{E}[A] + CI\)

  • \(b\): Insurer expense + loss control + premium audit G&A + Adj for limiting ratio + profit contingencies
  • \(e\): Total expense xTax and profit
  • \(I\): Net insurance charge for the max/min premium
  • Since \(e\) includes all expenses including LAE (except taxes), we subtract off the expected LAE, since we will instead use the LAE related to actual losses from the \(CA\) in the retro premium equation
  • Proof

2001 Q32

Understanding the Net Insurance Charge: w/o Net Insurance Charge

Ignore the retro development for now

\(R = (b + C \times A) \times T \: \: \: \text{for} \: H \leq R \leq G\)

Max and min premiums \(\equiv\) Aggregate limits on losses

Formula
2001 Q31

\(\begin{align} H &= (b + C \times A_H) \times T &= \text{min}\\ G &= (b + C \times A_G) \times T &= \text{max}\\ \end{align}\)

\(R = (b + C \times {\color{blue}L}) \times T \: \: \: \: {\color{blue}L} = \left\{ \begin{array} AA_H & A \leq A_H\\ A & A_H < A <A_G\\ A_G & A \geq A_G\\ \end{array} \right.\)

Understand Formula

\(\begin{array}{ccc} \text{Net Insurance Charge} = &I = (&\text{Ins Chg} &- &\text{Ins Saving}&) \times \mathrm{E}[A]\\ &I = (&\dfrac{\mathrm{E}[Loss > A_G]}{\mathrm{E}[A]} &- &\dfrac{\mathrm{E}[Loss < A_H]}{\mathrm{E}[A]}&) \times \mathrm{E}[A]\\ &I = (&\text{Tbl M Max Prem Chg} &- &\text{Tbl M Min Prem Savings}&) \times \mathrm{E}[A]\\ &I = (&\phi(r_G) &- &\psi(r_H)&) \times \mathrm{E}[A]\\ &I = (&\phi(\dfrac{A_G}{\mathrm{E}[A]}) &- &\psi(\dfrac{A_H}{\mathrm{E}[A]})&) \times \mathrm{E}[A]\\ \end{array}\)

Table M Terminology

Rows are entry ratios = \(r = \dfrac{Loss_{Actual}}{Loss_{Expected}}\) Understand

  • \(\because\) Standardized \(\therefore\) Table M can be applied to various states and LOB

Columns are for different risk size groups

  • Since distn of charges vary by risk size \(\Rightarrow\) Curve differ by risk size

Entry ratio at maximum premium \(= r_G = \dfrac{A_G}{\mathrm{E}[A]}\)

Entry ratio at minimum premium \(= r_H = \dfrac{A_H}{\mathrm{E}[A]}\)

Retrospective Rating: with Occurence Limit

Per occurrence limit impact the likelihood of hitting the max/min premiums \(\Rightarrow\) Table M not appropriate \(\Rightarrow\) Need to recognize the potential overlap between the occurrence and aggregate limits

3 options to handle this:

Option 1: Limited Loss Table M

Important Formulas

\(R^* = (b^{LLM} + CA^* + PCF + PCV) T\)Memorize

  • \(b^{LLM}\): From the limited loss Table M

  • \(A^*\): Reported limited loss @ t (might include ALAE)

  • \(F = ELF = \dfrac{\mathrm{E}[Loss \geq \text{Per Occ Limit}]}{\text{Standard Premium}}\)
    • Varies by State + HG + Occ Limit
    • Recall Robertson paper
  • \(A \neq A^* + PF\)

  • \(\mathrm{E}[A] = \mathrm{E}[A^*] = PF\)

  • \(PCF\) and \(PCV\) are electives

  • \(V\): Development different with occ limit than w/o


Theoretically correct approach

Caveat:

  • Requires a large number of tables

  • Charge need to vary by occ limit, entry ratio, risk size group, HG, etc

Option 2: Table M with ICRLL Adjustment (NCCI)

ICRLL = Insurance Charge Reflecting Loss Limitation

Change the Table M column used to be the column appropriate for a larger size risk

  • Per occurrence limit reduces the skewness of the entry ratio distn \(\Rightarrow\) Similarly for larger risk size

Caveat:

  • Table M doesn’t vary by loss limit \(\Rightarrow\) Compromises a bit on accuracy

Option 3: Table L (WCIRB)

\(R^* = (b^L + CA^* + PCV) T\)

Accurately correct for overlap between the occurrence and aggregate charges similar to the LLM

Combines occurrence charge with the aggregate charge

  • Whereas LLM’s occ charge has to be added separately

WCIRB uses different HG than NCCI

\(V\) varies by occ limit

  • \(\uparrow\) limits \(\uparrow\) \(V\) \(\because\) \(\uparrow\) potential for development

NCCI Filings for Optional Components

NCCI do full rates or loss costs depending on the state

Full Rates:

  • Files \(ELF\) from option 1

  • Files \(V\), Retro Development Factors, from option 1

Loss Costs:

  • Files \(ELPPF\) (XS Loss Pure Premium Factors)

    • \(\begin{array}{cccc} ELF &= &ELPPF &\times &ELR\\ \dfrac{\mathrm{E}[XSLoss]}{\text{Standard Premium}} &= &\dfrac{\mathrm{E}[XSLoss]}{\mathrm{E}[Loss]} &\times &\dfrac{\mathrm{E}[Loss]}{\text{Standard Premium}}\\ \end{array}\)

    • \(ELR\) from company’s own estimate

  • Use \(ELAEPPF\) if per occ limit includes Loss & ALAE

  • Files Development Retrospective Pure Premium Factors

    • \(\begin{array}{cccc} \text{Retro Dev Factor} &= &\text{Retro Dev PP Factor} &\times &ELR\\ \dfrac{\mathrm{E}[\text{Unreported Limited Loss}]}{\text{Standard Premium}} &= &\dfrac{\mathrm{E}[\text{Unreported Limited Loss}]}{\mathrm{E}[Loss]} &\times &\dfrac{\mathrm{E}[Loss]}{\text{Standard Premium}}\\ \end{array}\)

    • XS Loss and Allocated Expense Pure Premium Factors

Table M

Constructing Table M

Aggregate distn across a large number of risks

  • For losses, LR, entry ratios, etc
  • Each data point is a single policy for a single term

Table M Charge

Revisit! This seems wrong…

  • \(\begin{align} \text{Table M Charge} &= \dfrac{\sum_{r_i > r} Loss_i}{\sum_i Loss_i}\\ &= \dfrac{\text{Total Losses }\forall \text{ Policies w/ Losses > Entry Ratio r}}{\text{Total Losses }\forall \text{ Policies}} \end{align}\)

  • \(\begin{align} \text{Table M Savings} &= \dfrac{\sum_{r_i < r} Loss_i}{\sum_i Loss_i}\\ &= \dfrac{\text{Total Losses }\forall \text{ Policies w/ Losses < Entry Ratio r}}{\text{Total Losses }\forall \text{ Policies}} \end{align}\)

2 Components to build Table M

  • Sample of experience from a group of similarly sized risks
    • Risk measure can be actual losses, LR, etc
  • \(\mathrm{E}[\textit{Experience Measure}]\) from the sample

Normalize table if the sample average \(\neq\) assumed expected value:

  • Ignore actual expected \(\Rightarrow\) Use sample avg \(\Rightarrow\) \(\phi(0) = 1\)
  • Divide charge and entry ratio columns by \(\phi(0)\) to normalize

Constructing Table M Example

Annual Claim Total ($) # of Policies
1000 8
1500 3
4000 2
10000 5
15000 2
40000 1

Each risk has $10,000 of standard premium

\(\text{Max Premium} \Leftrightarrow\) 80% LR w.r.t. Standard Premium

\(\text{Min Premium} \Leftrightarrow\) 20% LR w.r.t. Standard Premium

Loss @ Max Premium = 80% \(\times\) $10,000 = $8,000 = \(A_G\)

Loss @ Min Premium = 20% \(\times\) $10,000 = $2,000 = \(A_H\)

Total # of Policies = 21

Total Losses = $140,500

\(\mathrm{E}[A] = \text{Expected Loss per Policy} = \dfrac{\text{Total Losses}}{\text{Total # of Policies}} =\) $6,690.48 = Total area under curve

# Transformation of the data to plot
d1_ecdf <-
  d1 %>%
  rbind(c(0,0),.) %>%
  mutate(`% Policies` = `# of Policies` / sum(`# of Policies`),
         `Claim EWCDF` = cumsum(`% Policies`))
# Note the addition of 0 row
Annual Claim Total ($) # of Policies % Policies Claim EWCDF
0 0 0.0000000 0.0000000
1000 8 0.3809524 0.3809524
1500 3 0.1428571 0.5238095
4000 2 0.0952381 0.6190476
10000 5 0.2380952 0.8571429
15000 2 0.0952381 0.9523810
40000 1 0.0476190 1.0000000

Note that the steps are drawn in a “vh” manner starting from origin

Table M Charge in dollars = Area under curve and above $8,000 line \(A_G\)

  • If divided by total area = Table M Charge

Table M Savings in dollars = Area under $8,000 line \(A_G\) and above curve

  • If divided by total area = Table M Savings

Net Insurance Charge = Difference of the Table M Charge in dollars and Table M Savings in dollars

3 Ways to Calculate the Areas

  1. Horizontal Slices (Table M Method)
    • Calculating the entry ratio first
  2. Vertical Slices
  3. Calculate area within limit and then subtract it from the expected loss
    • = Limiting losses at the max/min

Horizontal Slices (Table M Method)

Calculating the entry ratio first

  • Used to get the “height” of the segment

Starting from the top of the graph

Cumulate the horizontal length of each section from top to bottom

  • The % risks above \(r_i\) piece

Advantages:

  • Efficient when calculating multiple \(\phi{r_i}\)
  • Faster when there are many risks

Table M Construction from Sampled Loss

1. Start with table of sampled loss amount and respective frequency

Annual Claim Total ($) # of Policies
1000 8
1500 3
4000 2
10000 5
15000 2
40000 1

2. Add rows for 0 loss and entry ratio

tbl_m <-
  d1 %>%
  rbind(c(0,0), c(minlr * sp,0), c(maxlr *sp,0)) %>%
  arrange(`Annual Claim Total ($)`) %>%
  rename(`# risks` = `# of Policies`)

kable(tbl_m, align = 'l')
Annual Claim Total ($) # risks
0 0
1000 8
1500 3
2000 0
4000 2
8000 0
10000 5
15000 2
40000 1

3. Calculate Entry Ratio \(r\)

  • \(\dfrac{Loss_{Actual}}{Loss_{Expected}}\)
tbl_m <-
  tbl_m %>%
  mutate(`Entry Ratio (r)` = `Annual Claim Total ($)` / weighted.mean(`Annual Claim Total ($)`, `# risks`))

kable(tbl_m, align = 'l')
Annual Claim Total ($) # risks Entry Ratio (r)
0 0 0.0000000
1000 8 0.1494662
1500 3 0.2241993
2000 0 0.2989324
4000 2 0.5978648
8000 0 1.1957295
10000 5 1.4946619
15000 2 2.2419929
40000 1 5.9786477

4. Calculate # of risk above each row (Loss)

tbl_m <-
  tbl_m %>%
  mutate(`# risks above` = sum(`# risks`) - cumsum(`# risks`))

kable(tbl_m, align = 'l')
Annual Claim Total ($) # risks Entry Ratio (r) # risks above
0 0 0.0000000 21
1000 8 0.1494662 13
1500 3 0.2241993 10
2000 0 0.2989324 10
4000 2 0.5978648 8
8000 0 1.1957295 8
10000 5 1.4946619 3
15000 2 2.2419929 1
40000 1 5.9786477 0

5. Calculate % of risk above each row

tbl_m <-
  tbl_m %>%
  mutate(`% risks above` = `# risks above` / sum(`# risks`))

kable(tbl_m, align = 'l')
Annual Claim Total ($) # risks Entry Ratio (r) # risks above % risks above
0 0 0.0000000 21 1.0000000
1000 8 0.1494662 13 0.6190476
1500 3 0.2241993 10 0.4761905
2000 0 0.2989324 10 0.4761905
4000 2 0.5978648 8 0.3809524
8000 0 1.1957295 8 0.3809524
10000 5 1.4946619 3 0.1428571
15000 2 2.2419929 1 0.0476190
40000 1 5.9786477 0 0.0000000

6. Calculate the Table M Charge \(\phi(r)\)

\(\phi(r) = 0\) for the largest value of r \(\Rightarrow\) No \(\mathrm{E}[L]\) above the largest loss

\(\phi(r_i) = \phi(r_{i+1}) + (r_{i+1} - r_i)(\text{ % risks above }r_i)\) Memorize Formula

Savings \(= \psi(r) = \phi(r) + r - 1\) Memorize Formula

Vertical Slices

Looking at groups of losses based on the size of loss

Probably start from the right but don’t seem to matter

  • For Tbl M Charge: \(\text{ % of Policies hitting cap} \times \text{XS Loss Amount}\)
  • For Tbl M Savings: $

Advantages:

  • Natural, since that’s how the data is presented
  • Easier to understand
  • Faster if just looking at 1 \(r\)

Understanding Table M

Important Concepts


alt text

Note that the y axis is now the entry ratios and not losses like it was in the earlier example

  • This is like looking at a normal CDF but rotated along the x=y line

Defn:

\(Y = \dfrac{A}{\mathrm{E}[A]}\)

\(F(y) = \text{CDF of } y\)

Area under curve:

\(\mathrm{E}[Y] = \dfrac{\mathrm{E}[A]}{\mathrm{E}[A]} = 1\)

  • Recall when the y axis was loss amount, the area was \(\mathrm{E}[Y]\)

Table M Charge and Savings Formulas

\(\phi(r) = \int\limits_r^{\infty} (y - r) dF(y) = \int\limits_r^{\infty} (y - r) f(y)dy\)

\(\psi(r) = \int\limits_r^{\infty} (r - y) dF(y)\)

Key Properties - Tbl M Charge Important

\(\phi(0) = 1, \: \phi(\infty) = 0\)

\(\phi'(r) = -G(r) \leq 0\)

  • \(G(r) = 1 - F(r)\)

\(\phi''(r) = f(r)\)

Key Properties - Tbl M Savings Important

\(\psi(0) = 0, \: \psi(\infty) = \infty\)

\(\psi(r)'(f) = F(r) \geq 0\)

\(\psi''(r) = f(r) = \phi''(r)\)

Note from figure above, the area under \(r = \psi(r) + (1 - \phi(r)) \Rightarrow \psi(r) = \phi(r) + r -1\) style=“color:red;background-color:yellow”>Memorize

\(\phi\) is monotonically decreasing function of premium size

Table M Charges and Premium Size

Large sample and large risks \(\Rightarrow\) \(\downarrow\) variance in Loss Ratio and Entry Ratio than large sample but small risks

\(\lim\limits_{\text{Premium Size}\rightarrow \infty} Var(y) = 0\)

  • Curve will flatten out and look like all risks have the same amount of losses

\(\phi(r) = \begin{cases} 1 - r &r \leq 1\\ 0 &r > 1 \end{cases}\)

CDF of Entry Ratio vs Entry Ratio

alt text

From TIA: Think about how Table Ms are built: you get the loss ratios for a bunch of similar sized risks, calculate the average loss ratio from these risks, and then calculate entry ratios for each risk as actual loss ratio / average loss ratio. The variance in the entry ratios is really just the variance in the actual loss ratios of the risks used to built the Table M. If the variance of these loss ratios goes to 0, then that is equivalent to saying all risks have the same loss ratio (aka, same entry ratio, aka same dollar amount of aggregate losses given equal premium sizes). So that is why the curve will flatten.

On the other hand, as \(\lim\limits_{\text{Premium Size} \rightarrow 0} \phi(r) = 1\)

Graphs of Entry Ratios vs Table M Charge/Savings as premium size increase

  • Higher ELG (a different column) in Table M, which would have lower charges for a given entry ratio

alt text

Net Insurance Charge - w/o Occurence Limit

Important Concepts

\(R = (b + C \times {\color{blue}L}) \times T \: \: \: \: {\color{blue}L} = \left\{ \begin{array} AA_H & A \leq A_H\\ A & A_H < A <A_G\\ A_G & A \geq A_G\\ \end{array} \right.\)

\(A_H = r_H \mathrm{E}[A]\) and similarly for \(A_G\)


alt text

Blue area = \(\dfrac{\mathrm{E}[{\color{blue}L}]}{\mathrm{E}[A]}\)

  • Recall that \(\mathrm{E}[{\color{blue}L}]\) is the expected limited loss
  • = Expected Retro Premium? 2003 Q29

Memorize Formula

\(\begin{array}{ll} \dfrac{\mathrm{E}[L]}{\mathrm{E}[A]} &= 1 + \psi(r_H) - \phi(r_G) \\ \mathrm{E}[L] &= \mathrm{E}[A] + [\psi(r_H) - \phi(r_G)]\mathrm{E}[A] \\ \mathrm{E}[L] &= \mathrm{E}[A] - I \end{array}\)

  • Recall: \(I = \mathrm{E}[A][\phi(r_G) - \psi(r_H)]\)
  • Expected loss given max/min = Expected Loss - Net Insurance Charge

Memorize Understand

\(\begin{array}{lll} R &= (b + CL)T\\ \mathrm{E}[R] &= (b + C\mathrm{E}[L])T\\ \mathrm{E}[R] &= \big\{b + C(\mathrm{E}[A] - I)\big\}T & (1)\\ \end{array}\)

Recall if the plan is balance \(\Rightarrow\) \(\mathrm{E}[R] = \text{Guaranteed Cost Premium}\)

\(\begin{array}{lll} \text{Guaranteed Cost Premium} &= (e + \mathrm{E}[A])T & (2)\\ &=(1-D)P & (3)\\ \end{array}\)

  • \(e\) = Total expense x tax & profit
  • \(D\) = Premium discount \(= \sum (\text{Premium in Range}_i)(\text{Discount}_i)\)
  • \(P\) = Standard Premiums

Combine \((1)\) and \((2)\)

\(\big\{b + C(\mathrm{E}[A] - I)\big\}T = (e + \mathrm{E}[A])T\)

\(b = e - (C - 1)\mathrm{E}[A] + CI\) Memorize Formula

  • \(e - (C - 1)\mathrm{E}[A]\) is never capped
  • \(CI\) is where the cap comes in
  • Recall: Check \(e = \dfrac{1-D}{T} - PLR\)
    \(T = \dfrac{0.2 + PLR(1+\mu)}{0.2 + PLR}\left(\dfrac{1}{1-\tau}\right)\)

2009 Q31, 2011 Q8

Question that ask for G or H usually use balance equation

Alt formula: \(b = (1 - D)\dfrac{P}{T} - C\mathrm{E}[A] + CI\)

Use this if we don’t’ have max/min premium

Table M Balance Equations

Basic premium and max/min premium depends on each other \(\Rightarrow\) Need trial and error to get the right Table M row

First balance equation

Value (change) difference

Retro premium formula with max loss:
\(H = (b + CA_H)T = (b + Cr_H\mathrm{E}[A])T\)

Start with the GCP formula and subtract \(H\)

\(\begin{array}{ll} \text{GCP} - {\color{blue}H} &\\ (e + \mathrm{E}[A])T - {\color{blue}H} &= \big\{b + C(\mathrm{E}[A] - I)\big\}T - {\color{blue}{(b + Cr_H\mathrm{E}[A])T}}\\ &= CT\big\{\mathrm{E}[A] - I - r_H\mathrm{E}[A]\big\}\\ &= CT\Big\{\mathrm{E}[A] - \mathrm{E}[A]\big\{\phi(r_G) - \psi(r_H)\big\} - r_H \mathrm{E}[A]\Big\}\\ &= C \: T \: \mathrm{E}[A] \big[ \psi(r_H) - r_H + 1 - \phi(r_G)\big] \\ \end{array}\)

Move everything to one side and we get: Important Memorize

\(\phi(r_H) - \phi(r_G) = \dfrac{(e + \mathrm{E}[A])T - H}{C\mathrm{E}[A]T} = \dfrac{(1-D)P-H}{C\mathrm{E}[A]T}\)

  • Interpretation: Charge difference = to the difference between the expected retro premium E[R] and the minimum premium H, except for the factor CE[A]T
  • 2003 Q29

Second balance equation

Entry difference, based \(G - H\)

Important Memorize

\(r_G - r_H = \dfrac{G - H}{C\mathrm{E}[A]T}\)

Can do all the above looking at them as a % of Standard Premium

Limited Loss Table M

Constructing a Limited Loss Table M

Same as Table M but with limited losses \(A^*\) for each policy

Entry Ratio: \(r^* = \dfrac{A^*}{\mathrm{E}[A^*]}\)

Note that the denominator of the Entry Ratio is the \(\mathrm{E}[A^*]\), where as in Table L it’s not limited \(\mathrm{E}[A^*]\)

Understanding a Limited Loss Table M

Note that the y axis is now limited entry ratios \(y^*\)


alt text

Entry Ratio: \(Y^* = \dfrac{A^*}{\mathrm{E}[A^*]}\)

\(F^*(Y^*) = \text{CDF of }Y^*\)

Loss Elimination Ratio: \(k = LER = 1 - \dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A]}\) Know

  • \(\dfrac{\mathrm{E}[A^*]}{P} = \text{Unlimited ELR} - LER\)

Note that \(\mathrm{E}[Y^*] = \dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A^*]}\) no surprise

But \(\mathrm{E}[Y] = \dfrac{\mathrm{E}[A]}{\mathrm{E}[A^*]} = 1 + k\dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A]}\)

Integration and key properties all the same as Table M but with limited loss

Same thing for what happens when \(\lim\limits_{\text{Premium Size} \rightarrow \infty}\) and \(\lim\limits_{\text{Premium Size} \rightarrow 0}\)

Net Insurance Charge - LLM

\(R^* = (b^{LLM} + C{\color{blue}{L^*}}+PCF) T \: \: \: \: {\color{blue}{L^*}} = \left\{ \begin{array} AA_H & A^* \leq A_H\\ A^* & A_H < A^* <A_G\\ A_G & A^* \geq A_G\\ \end{array} \right.\)

\(+ PCV\) if to stabilize

Everything similar to Table M but now that the entry ratios \(r^*\) are limited and has some limied loss \(\mathrm{E}[A^*]\)

\(\mathrm{E}[L^*] = \mathrm{E}[A^*] - I^{LLM}\)

\(b^{LLM} = e - (C - 1)\mathrm{E}[A] + CI^{LLM}\)

  • \(e - (C - 1)\mathrm{E}[A]\) is never capped

\(\phi^{LLM}(r_H^*) - \phi^{LLM}(r_G^*) = \dfrac{(e + \mathrm{E}[A])T - H}{C\mathrm{E}[A^*]T} = ? \dfrac{(1-D)P-H}{C\mathrm{E}[A^*]T}\)

\(r_G^* - r_H^* = \dfrac{G - H}{C\mathrm{E}[A^*]T}\)

Table L

Constructing Table L

Table L charge includes the charge for per occurrence limit

  • LLM adds the per occ charge via ELF separately (by State + HG + occ limit)

Implicit charge for the occ limit \(\in \{0, k\}\)

  • \(k\) is the Loss Elimination Ratio
  • Incremental charge above the normal Table M charge.
  • Varies by: occurrence limit, ELR, premium size, entry ratios
    • Not vary by premium size would result in overcharging small risks and undercharging large risk

Table L uses different premium size groups and occurrence limits

The LER was derived from the combination of all premium groups

  • That’s why entry ratio are based on expected unlimited losses \(\mathrm{E}[A]\)

CA version of Table M shows a higher charge than the CW average due to higher variation in the CA rates

\(\phi^L(r) - \phi(r)\) shows in the data was smaller than the LER implies due to overlap between occ limit and agg limit

Main Difference from the M and LLM

Entry ratios = \(\dfrac{A^*}{\mathrm{E}[A]} = \dfrac{\text{Actual Limited Loss}}{\text{Expected }\textbf{Unlimited}\text{ Loss}}\)

2013 Q15b

If Standard Premium \(\neq\) \(\forall\) risk \(\Rightarrow\) Create Tbl L with P instead of # of risk

  • Denominator of Entry Ratios \(= \dfrac{\mathrm{E}[A]}{P}\)
  • Numerator of Entry Ratios \(= \dfrac{A^*}{P}\)
  • \(\phi^L(r_i) = \phi^L(r_{i+1}) + (r_{i+1} - r_i)(\text{ % Premium above }r_i)\)

2008 Q32

Understanding Table L


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Note that the y axis is \(Y = \dfrac{A^*}{\mathrm{E}[A]}\) and the CDF of interest is \(F^*(Y)\). However, the x axis of the graph is of \(F(Y)\)

  • Axis = Original Table M since denominator of the Entry Ratio \(= \mathrm{E}[A]\)

Loss Elimination Ratio: \(k = LER = 1 - \dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A]}\)

  • Same as LLM

Area under \(F(Y) = 1\) same as Table M \(\Rightarrow\) Area under \(F^*(Y) = \mathrm{E}[Y] = \dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A]} = 1 - k\)

  • Based on graph

\(\phi^L(r) = \int\limits_r^{\infty} (y-r)dF^*(y) + k\) Remember the k

  • Note the addition of k, charge for per occ limit, is in the table L charge

\(\psi^L(r) = \int\limits_0^{r} (r-y)dF^*(y)\)

Key properties all the same except:

\(\phi^L(\infty) = k\)

Net Insurance Charge - Table L

\(R^* = (b^L + C{\color{blue}{L^*}}) T \: \: \: \: {\color{blue}{L^*}} = \left\{ \begin{array} AA_H & A^* \leq A_H\\ A^* & A_H < A^* <A_G\\ A_G & A^* \geq A_G\\ \end{array} \right.\)

Everything same as Table M but now that we use \(I^L\)

\(\mathrm{E}[L^*] = \mathrm{E}[A^*] - I^{L}\)

\(b^{L} = e - (C - 1)\mathrm{E}[A] + CI^{L}\) * \(e - (C - 1)\mathrm{E}[A]\) is never capped

\(\phi^{L}(r_H) - \phi^{L}(r_G) = \dfrac{(e + \mathrm{E}[A])T - H}{C\mathrm{E}[A]T} = ? \dfrac{(1-D)P-H}{C\mathrm{E}[A]T}\)

\(r_G - r_H = \dfrac{G - H}{C\mathrm{E}[A]T}\)

2001 Q34

Table L vs NCCI

NCCI uses the ICRLL to \(\approx\) LLM

Table L Advantages:

  • Mathematically accurate in how it accounts for the overlap
  • No need for separate per occ charge (\(PCF\) in LLM)
  • CA focus

Table L Disadvantages:

  • Can’t be used for alternate loss limits
    • Charge for a fixed limit is built into the table
  • Requires large numbers of tables for CW
    • Varies by Entry Ratio + State + Limit + Premium Size Group
    • Likely by HG too since ELFs varies by HG
  • Need regular update for inflation
    • Aggregate distn \(\Delta\) shape
    • With Table M you just update the \(\mathrm{E}[L]\)
  • CW data might be more credible

Example of Table L, LLM, and M

See Manual for example

Using NCCI Manual

Some of the important sections below. Cancellation provision is not on the syllabus

Expected Loss Group Tables (Pt 4-B)

Expected Loss Group = Risk Size Groups

To determine the ELG:

  1. Adjusted Expected Losses \(= \mathrm{E}[A] \times \text{State HG Differential} \times \dfrac{1 + 0.8k}{1-k}\)
    • \(k = LER\)
    • \(\dfrac{1 + 0.8k}{1-k}\) is the ICRLL procedure
      • LLM \(\approx\) Table M by adjusting the Adjusted Expected Losses to be for a ELG
      • Use LLM balance equation instead if using this
  2. Look up ELG with the Adjusted Expected Losses
    • Match the policy effective date with table effective date

\(\mathrm{E}[L]\) increase over time due to inflation. The ELG tables are updated so the curves so NCCI doesn’t have to update the curve. They’ll just push risk to different ELG.

Estimate portion of risk moving from ELG X to Y: \(\dfrac{\text{Portion of ELG X in new ELG Y}}{\text{Size of old ELG X}}\)

Since there is more variance in the loss ratio distn for smaller insureds than for larger insureds, there will be more variance in the entry ratio distn as well. As such, the tables in the NCCI manual reflect different columns for different insured sizes. Using a single distn for all insureds would result in overcharging the large insureds and undercharging the small insureds

Table M (Pt 4-C)

Each column is for different ELG

For low entry ratios, Table M savings are shown as well

Expense Ratio Tables (Pt 4-D)

Not tested since 2000

This is for the \(e\) in \(b = e - (C - 1)\mathrm{E}[A] + CI\)

Type A = Stock; Type B = non-Stock

Shows ELR, premium discount ranges and tax multipliers

AK Special Rating Values

2nd last page

  • Items to get Adjusted Expected Losses
    • HG Differentials; varies by HG
    • ELPPF by occ limit + HG
      • ELPPFs = LER \(k\)

Last page

  • ELAEPPFs
  • Retrospective PP Dev Factor
    • By occ limit and adjustments
    • Can convert into retrospective development factor \(V\)

Cliff’s Summary

Lee 2

Know how the entry ratio, charges and other relationships are represented as areas in the graph

Different forms of the charge and properties

Gillam & Snader 2

Using NCCI Table M

  • Need Adj Expected Loss to get ELG to get know the column of Tbl M

Remember the basic premium formula only works if plan is balanced

  • \(b = e - (C - 1)\mathrm{E}[A] + CI\)

Know how to build Table M

Know how to plug and play with all the equations

Brosius

Vertical and horizontal calculations of the tables

  • Remember how to normalize

Table construction

Skurnick

Table L rationale pros cons

ICRLL rationale pros cons

Unbalanced retro calculation

Insurance charge and savings for Table L

NCCI 2

How to use the NCCI pages

  • ELG, balance equations
  • Use the table for the right year
  • Look for entry difference first then charge

Types of Exam Questions

Lee 2

Interpret entry ratio plots

  • 2000 Q19 (Interpret)
  • 2001 Q30 (Explain its constructions)
    • Understand that Table M is constructed horizontally

How charges are represented on graphs

  • 2002 Q17
  • 2003 Q8
  • 2003 Q29 (interpret charge difference)
    • ?Question?
  • 2007 Q31 (both L and M)

Draw graphs and calculate charges

  • 2005 Q33 (uniform)
  • 2006 Q29 \(\star\)
  • 2008 Q29 (uniform)
  • 2009 Q9 \(\star\) (Calc expected retro premium)
  • 2010 Q25
  • 2012 Q21
    • Convert \(r\) to LR to do this?

Basic premium from formula

  • 2012 Q23 \(\star\)

Impact of change on losses or size

  • 2012 Q21 b TIA
  • 2013 Q15 b,c \(\star\)

Other

  • 2002 Q14 (XS PP formula)
    • Why is the denominator 1??Question?

Gillam & Snader 2

Using Table M

  • 2000 Q5 (ELG)

Plug and play for \(b\), \(R\), \(GCP\), etc

  • 2000 Q6 (Know how to do premium discount)
  • 2000 Q9 (\(GCP\) and \(A\))
  • 2001 Q8 (\(GCP\))
  • 2001 Q31 (\(b\) formula and ELG)
    • Not sure about part b
    • TIA
  • 2001 Q33 (Retro @18mo) \(\star\)
  • 2002 Q15 (Charge difference)
  • 2002 Q16 (net converted charge)
  • 2002 Q40a (Retro @18mo)
  • 2003 Q7 (Tax multiplier)
  • 2003 Q10 (\(b\))
  • 2004 Q18 (Retro @18mo)
  • 2004 Q20 (Back out \(e\) with GCP and retro)
  • 2005 Q30a (Tax multiplier)
  • 2005 Q30b (Tax multiplier w/ interstate risk)
  • 2005 Q32a (GCP and Retro Prem)
  • 2006 Q30 \(\star\) (Retro, hit cap)
  • 2006 Q32 \(\star\) (max prem factor, clever)
  • 2010 Q27 \(\star\) (diff between limited and unlimited)
  • 2010 Q29 \(\star\) (Multiple items)
  • 2011 Q20 \(\star\) (GCP, expected loss is tricky)
  • 2011 Q25 \(\star\) (max prem, need to 2 balance equation)
  • 2012 Q19 \(\star\) (Retro @ 2nd adjustment)
  • 2013 Q14 \(\star\) (expected retro, need that E[L]/E[A] formula)
  • 2014 Q18 \(\star\)\(\star\)

Formula Manipulation

  • 2001 Q32a \(\star\)

Build Table M

  • 2000 Q48

Concepts

  • 2001 Q10 (relationship between max/min/b)
    • Do we need the assumed balance condition here?
  • 2001 Q32b (iterative tbl M lookup)
  • 2002 Q40b \(\star\) (Improve insured cashflow)
  • 2003 Q33 \(\star\) (\(\Delta\) in exposure impacts)
    • Not sure about b
  • 2004 Q47 \(\star\) (How to lower b)
  • 2005 Q32a (GCP and Retro Prem difference)

Tricky

  • 2007 Q32 (defective)
  • 2007 Q35 (Balanced or not)
  • 2009 Q20 (defective)

Brosius

Insurance charge/savings

  • 2002 Q36 (vertical)
  • 2004 Q43 (vertical, tricky arithmetics)
  • 2005 Q8 (vertical, charge/savings relationship)
  • 2006 Q9 (table construction)
  • 2007 Q30 (vertical)
  • 2007 Q34 (table construction)
  • 2008 Q28b (table construction)
  • 2013 Q12 (table construction)
  • 2014 Q13 \(\star\) (Backout numbers)
  • Practice \(\star\) (table construction with normalize)

Concepts

  • 2008 Q28a (vertical vs horizontal)

Skurnick

Understandig Table L and ICRLL

  • 2000 Q45 \(\star\) (Tbl L and ICRLL, Pros Cons)
  • 2002 Q38
  • 2006 Q34 \(\star\) (per occ charge)
  • 2008 Q35 (same as 2000 Q45)
  • 2010 Q31 \(\star\) (ICRLL proc)
  • 2011 Q22 bc
  • 2013 Q13 \(\star\) (Relationship with prem size and properties)
  • 2014 Q12 a \(\star\) (build table from graph)
  • 2014 Q12 b \(\star\) (Explain savings)

Unbalanced retro calc

  • 2001 Q34 \(\star\) (Unbalance and then rebalance)
    • Good questions to understand

Table Construction

  • 2003 Q30 (L and M)
  • 2009 Q32 (L and M)
  • 2011 Q22a (L)
  • 2012 Q18 \(\star\)

Calculate insurance charge/savings

  • 2004 Q44 (Vertical, L and M)
  • 2006 Q7 (Vertical, L)
    • Why is the per-occ charge not always just the LER??Question?
  • 2008 Q32 \(\star\star\) (Table L with different SP risk)
    • Use dollar amount or Tbl L with premium (instead of risk)
    • Should understand this

Must Know Formulas

NCCI 2

Plug and play

  • 2001 Q9 (LLM charge difference)
  • 2002 Q35 \(\star\)
    1. expense and contingency in \(b\) = w/o the \(CI\)
    2. \(CI\) (Get ELG with LER + balance equation \(\rightarrow\) tbl M search)
    3. Total
  • 2003 Q31 (ELG and balance equations and D)
  • 2004 Q45 (similar to 2002 Q35)
  • 2005 Q32 (Just charge)
  • 2006 Q35 \(\star\) (backout ELF)
  • 2008 Q36 (similar to 2002 Q35)
  • 2009 Q30 (similar to 2002 Q35)
  • 2011 Q21 \(\star\) (Alaska ELG)
    • Lookup HG differential and ELAEPPF
  • 2014 Q15 \(\star\) (inflation impact)
  • 2014 Q17 \(\star\)
    1. GCP (Tax is applied to final P)
    2. \(b^{LLM}\)
    3. Impact of higher per occ limit

Concepts

  • 2003 Q32 (Why varies by ELG)

Table M

Net Charge (For LLM use \(\mathrm{E}[A^*]\) and LLM charges):

  • \(I = \mathrm{E}[A][\phi(r_G) - \psi(r_H)]\)

Savings:

  • \(\psi(r) = \phi(r) + r -1\)

Other Relationships:

  • \(\begin{array}{ll} \mathrm{E}[L] &= \mathrm{E}[A] \left\{1 + \psi(r_H) - \phi(r_G)\right\} \\ &= \mathrm{E}[A] - I \\ \end{array}\)

Table construction:

  • \(\phi(r_i) = \phi(r_{i+1}) + (r_{i+1} - r_i)(\text{ % risks above }r_i)\)

Reading Table M

Balance Equations (Use \(\mathrm{E}[A^*]\) for the denominator if there’s occ limit):

  • \(\phi(r_H) - \phi(r_G) = \dfrac{(e + \mathrm{E}[A])T - H}{C\mathrm{E}[A]T}\)

  • \(r_G - r_H = \dfrac{G - H}{C\mathrm{E}[A]T}\)

Adjusted Expected Losses:

  • \(= \mathrm{E}[A] \times \text{State HG Differential} \times \dfrac{1 + 0.8k}{1-k}\)

Questions

What is the x-axis for the Table LLM

What is the state differential in finding the Adjusted Expected Loss

What is that E[L]/E[A] formula

About This

Just a bunch of notes for Exam 8

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